# ELASTICITY (MDM 10203)

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1 LASTICITY (MDM 10203) Lecture Module 5: 3D Constitutive Relations Dr. Waluyo Adi Siswanto University Tun Hussein Onn Malaysia

2 Generalised Hooke's Law In one dimensional system: = (basic Hooke's law) Considering the axis: x = x x When looking the Poisson's ratio (Lecture Module 3), it has been shown the relation of the strain and the stress in principle axes: { x 1 y }= [ 1 x 1 1 ]{} vector vector matrix MDM Dr. Waluyo Adi Siswanto 2

3 Generalised Hooke's Law { x 1 y }= [ 1 x 1 1 ]{} which implies }=[ { x [ ]] 1 1 } { x vector (3x1) matrix (3x3) vector (3x1) MDM Dr. Waluyo Adi Siswanto 3

4 Generalised Hooke's Law When it is extended to all stresses and strains (9 components of the tensors): { x x}=[ ]{ x x} vector (6x1) matrix (6x6) vector (6x1) MDM Dr. Waluyo Adi Siswanto 4

5 Generalised Hooke's Law Constitutive Matrix { x D 21 D 22 D 23 D 24 D 25 D 26 D 31 D 32 D 33 D 34 D 35 D 36 D 41 D 42 D 43 D 44 D 45 D 46 D 61 D 62 D 63 D 64 D 65 D 66]{ x D11 D12 D13 D14 D15 D16 x}=[ D 51 D 52 D 53 D 54 D 55 D 56 x} { }= [ D ] { } Constitutive matrix or lasticity matrix MDM Dr. Waluyo Adi Siswanto 5

6 Generalised Hooke's Law Compliance Matrix [C ]=[ D] 1 { x C 11 C 12 C 13 C 14 C 15 C 16 x C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36 x}=[ C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66]{ x} { }= [C ] { } Compliance matrix MDM Dr. Waluyo Adi Siswanto 6

7 Generalised Hooke's Law Anisotropic very direction has different properties, but D ij =D ji There are only 21 independent elastic constants in the generalised constitutive (Hooke's) law. { x D 21 D 22 D 23 D 24 D 25 D 26 D 31 D 32 D 33 D 34 D 35 D 36 D 41 D 42 D 43 D 44 D 45 D 46 D 61 D 62 D 63 D 64 D 65 D 66]{ x D11 D12 D13 D14 D15 D16 x}=[ D 51 D 52 D 53 D 54 D 55 D 56 x} MDM Dr. Waluyo Adi Siswanto 7

8 Anisotropic with one plane elastic symmetry x ' z ' z y ' x x-y plane (rotate 180deg about z) [Q ]=[ ] y [ ' ]=[ xx xy xz x y 0 0 1][ x y z ][ ] =[ [ ' ]=[ xy xz x y 0 0 1][ xx x y z][ ] =[ xx xx xy ] xz x y x y z ] xy xz x y x y z MDM Dr. Waluyo Adi Siswanto 8

9 Anisotropic with one plane elastic symmetry It can be written { x x D 21 D 22 D 23 D 24 D 25 D 26 D 31 D 32 D 33 D 34 D 35 D 36 D 41 D 42 D 43 D 44 D 45 D 46 D 61 D 62 D 63 D 64 D 65 D 66]{ D11 D12 D13 D14 D15 D16 x}=[ D 51 D 52 D 53 D 54 D 55 D 56 x} { x D11 D12 D13 D14 D15 D16 D 21 D 22 D 23 D 24 D 25 D 26 D 31 D 32 D 33 D 34 D 35 D 36 x}=[ D 41 D 42 D 43 D 44 D 45 D 46 D 51 D 52 D 53 D 54 D 55 D 56 ]{ x yz D 61 D 62 D 63 D 64 D 65 D 66 x} MDM Dr. Waluyo Adi Siswanto 9

10 Anisotropic with one plane elastic symmetry As a result: { x D11 D12 D13 D D 21 D 22 D 23 D D 31 D 32 D 33 D x}=[ D 41 D 42 D 43 D D 55 D D 65 D 66]{ x x} There are 13 independent elastic constants MDM Dr. Waluyo Adi Siswanto 10

11 xample Problem 5-1 Map the Constitutive matrix of anisotropic material If the symmetrical plane is y-z MDM Dr. Waluyo Adi Siswanto 11

12 xample Problem 5-2 Map the Constitutive matrix of anisotropic material If the symmetrical plane is z-x MDM Dr. Waluyo Adi Siswanto 12

13 z y ' Orthotropic (Anisotropic with three plane elastic symmetry) x ' x [Q ]=[ ] y z ' [ ' ]=[ xy xz x y 0 0 1][ xx x y z][ ] =[ xx xy ] xz x y x y z [ ' ]=[ xy xz x y 0 0 1][ xx x y z ][ ] =[ ] xx xy xz x y x y z MDM Dr. Waluyo Adi Siswanto 13

14 Orthotropic (Anisotropic with three plane elastic symmetry) { x D11 D12 D D 21 D 22 D D 31 D 32 D x}=[ D D D 66]{ x x} There are 9 independent elastic constants MDM Dr. Waluyo Adi Siswanto 14

15 Isotropic lasticity In isotropic material, the elasticity (modulus of elasticity) behaves similarly in any direction = x x = = Considering Poisson's ratio x = x = x = x G= 2 1 and shear strains or = G = 2G = G = 2G xz = xz G xz = xz 2G MDM Dr. Waluyo Adi Siswanto 15

16 Isotropic lasticity then the equation can be written in a single matrix equation: { x x}= [ x ]{ x} [C ] MDM Dr. Waluyo Adi Siswanto 16

17 Isotropic lasticity then the equation can be written in a single matrix equation: { x x}= [ x ]{ x} [C ] MDM Dr. Waluyo Adi Siswanto 17

18 Isotropic lasticity { }=[C ]{ } { }=[C ] 1 { } { x x}= { }=[ D]{ } [ ]{ x x} [ D ] MDM Dr. Waluyo Adi Siswanto 18

19 Isotropic lasticity { x x}= [ ]{ x x} [ D ] MDM Dr. Waluyo Adi Siswanto 19

20 xercise Problem 5-3 In tensor notation, ij = kk ij 2G ij ij = 1 ij kk ij Write in full matrix notation MDM Dr. Waluyo Adi Siswanto 20

21 ij = kk ij 2G ij xx = xx y z 2G xx y = xx y z 2G y z = xx y z 2G z =2G =2G x =2G x In matrix, the same with that in page 19 MDM Dr. Waluyo Adi Siswanto 21

22 ij = 1 ij kk ij xx = 1 y = 1 z = 1 = 1 = 1 x = 1 xx xx y z y xx y z z xx y z x In matrix, the same with that in page 17 MDM Dr. Waluyo Adi Siswanto 22

23 xample Problem 5-4 The component of the strain tensor at a point in a body are given by x =0.005, =0.004, = =0.001, =0.0005, x =0.002 If the modulus of elasticity = N / mm 2 and the Poisson's ratio =0.25 a) Determine the component of stress tensor. b) Write the codes in Freemat so that you can use for future calculation with different variables. MDM Dr. Waluyo Adi Siswanto 23

24 Orthotropic lasticity There are three moduli of elasticity: x y z There are three moduli of rigidity: G xy G yz G zx There are six Poisson's ratio: = x, =, x = x x = x, y =, xz = x MDM Dr. Waluyo Adi Siswanto 24

25 Orthotropic lasticity { x x}=[ 1 x y x x 1 y x x xz z z y y 1 z G xy G yz 0 1 G zx ] { x x} MDM Dr. Waluyo Adi Siswanto 25

26 xample Problem 5-5 Write Freemat codes to calculate stress tensor of orthotropic material with 12 independent variables as written in page 19. MDM Dr. Waluyo Adi Siswanto 26

27 Strain nergy Density Function In simple axial problem Strain energy is calculated by the area of proportional area 1 2 pl pl 1 2 pl pl In matrix system to obtain scalar U = 1 2 { }T [ D]{ } This is the Strain nergy Density Function MDM Dr. Waluyo Adi Siswanto 27

28 Thermoelastic Constitutive Law The total strain consists of two component: mechanical and thermal M ij = T ij ij M ij = 1 ij kk ij T ij = T T o ij ij = 1 ij kk ij T T o ij ij = kk ij 2G ij 3 2G T T o ij Duhamel-Neumann thermoelastic constitutive law MDM Dr. Waluyo Adi Siswanto 28

29 MDM Dr. Waluyo Adi Siswanto 29

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